Even if Taringail remained a Damodred, why on Earth would his Royal born children NOT be Mantears. If Galad was "merely attached to Mantear" he would not be a viable choice as First Prince of the Sword for a potential Sister produced from the union. There is no reason a Daughter Hier would forfeit her own childrens birthright to marry someone below her station... maybe a second son would be treated as such. By marrying Taringale she effectively removed him from Laman's line of succession ... as we know from later events ... So the Royal Daughter would have held the political advantage to the declining Damodred line.

Given that Andorans also do no preclude males from assuming the role of a High Seat it also makes no sense to willingly exclude Galad from succession to a powerful Andoran House ... Taringail was not High Seat of Damodred at Galad's birth (Laman was still alive) and with several other candidates (Laman's brothers) in line before him it seems like a very poor decision to exclude his male children from better inheritance in an Andoran Royal house, I assume there is no argument that any Daughter of Manter he may have produced would have been a Mantear. As you stated, Taringail would have been eliminated from the Cairhienin throne upon marrying a foreign Queen (Daughter Heir) which was well before the death of Laman, he was likely never a candidate for High Seat of Damodred for the same reason ... When Laman died another became High Seat despite Taringail's being alive and in a powerful position. And interestingly enough Galad was not made High Seat at that time either.

Perhaps Galad reverted to a Damodred due to Morgase's succession and subsequent marriage to Taringail in order to keep him from being a political tool in Andor. Even still his claim to the Sun Throne couldn't have been particularly strong since he is not a High Seat of any house. Elayne's is tenuous at best and only better because she has the support of the High Seat of Damodred thanks to Rand. Galad would be no better than a 3rd choice under normal circumstances (Toram and Caraline being top choices) and probably well down the line of succession given his odd situation. Perhaps if he was a politically driven man he could have made more of his Cairhienin heritage, but he has been pushed aside and is likely viewed by the Cairhienin as an Andoran despite the Damodred name.